Electrical differentiator



Aug. 2, 1960 C. J. HIRSCH ELECTRICAL DIFFERENTIATOR Filed Oct. 15, 1956 TRIGGERED k [E(i )E u p] or kAE m |0 3SOURCE OF VARIABLE a, VOLTAGE l o DIRECT- 0 CURRENT OAMPLIFIER L Eu )-E (1 mam ELECTRICAL DIFFERENTIATOR Charles J. Hirsch, Locust Valley, N .Y., assignor to Hazeltine Research, Inc., Chicago, Ill., a corporation of Illinois a r Filed Oct. 15, 1956, Ser. No. 616,057

17 Claims. (Cl. 235-183) General This invention relates to electrical computers for the Stats Patent circuit diagram of an electrical diiferentiator.

2,947,480 Patented Aug. 2,' 1960 ice Description of Fig. 1 diflerenliator Referring to Fig. 1 of the drawing, there is shown a The differentiator comprises circuit means for supplying a first voltage representing a function to be differentiated. Such means is represented by a source of variable voltage 10 and may comprise one of several simple electrical or electrical-mechanical arrangements. One simple arrangement would comprise a device for tracing the outline of a curve representing the variation of the variable parameter and a simple potentiometer and battery circuit for converting such variations into varying voltages. The diiferentiator also includes a storage device. This storage device may comprise, for example, condenser 13. Terminal 15, of

, condenser 13 is connected to terminal 12 of the source of solution of equations involving the operation of differentiation. The subject matter of this application is related to that of applicants Patents Nos. 2,652,194, 2,666,576,

and 2,671,608, all entitled Electrical Computer," and to a i copending application Serial No. 616,056, entitled Elec The operation of such analogue computers is usually continuous in nature so that 1 such computers are useful with instruments, such as tachometer instruments, which develop a continuous type signal. Continuous type analogue computers .heretofore employed to perform the mathematical operationof differentiation generally resort to direct-current feedback amplifiers having properly proportioned resistor-condenser networks in the feedback circuits and are further characterized by extremely high gains and large quantities of feedback. Such high-gain amplifiers, however, are relatively expensive and are subject to various disturbances.

The subject matter of applicants Patent No. 2,671,608 relates to solving problems involving differentiation. The present invention is an improvement thereover and performs the foregoing operation in a simpler fashion and utilizes fewer and less complex components.

It is an object of the present invention, therefore, to provide a new and improved electrical ditferentiator which avoids one or more ofthe limitations and disadvantages of prior difierentiators.

It is a further object of the invention to provide a new and improved electrical diiferentiator which does not require mechanical moving parts, is compact and light in weight, yet is capable of making computations involving operations of differentiation at high speeds.

In accordance with the invention, an electrical diflieren tiator comprises circuit means for supplying a first volt age representing a function to be differentiated. The differentiator includes a storage device and a circuit for substantially periodically and instantaneously applying the voltage to the storage device. The difi'erentiatorfurther includes a circuit for determining the difference between the instantaneous value of thefirst voltage and the voltage of the storage device and, in conclusion, the differentiator includes means for utilizing the difference as a representation of the derivative of the function.

For a better understanding of the present invention, together with other and further objects thereof, reference is had to the following description taken in connection with the accompanying drawing, and its scope will be pointed out in the appended claims.

Referring to the drawing: 8 j

Fig. 1 is a circuit diagram, partly schematic, of ajrepresentative embodiment of an electrical dilferentiator em bodying the present invention-in a particular form, and

Figs. 2a and 2b representfcurves useful in explaining the operation of the Fig. 1 embodiment.

variable voltage 10.

Additionally, the, diiferentiator includes a circuit for periodically and instantaneously applying said voltage to the storage device. This circuit may comprise a sinewave generator 17 coupled to a transformer 18. The output of transformer 18 is coupled to comparison circuit 21. Comparison circuit 21 is, in turn, coupled to a triggered pulse generator 31 having an output to sampling circuit 32. The output of sampling circuit 32 is coupled through terminal 14 of condenser 13 to a smoothing circuit 40. The output terminals of smoothing circuit 40 are coupled to direct-current amplifier 39.

The output of sine-wave generator 17 is connected to the primary winding of transformer 18 for translating a sinusoidal signal to the remaining circuits in the difien entiaton. Transformer 18 represents a simple means, of

many possible means, for supplying the sinusoidal signal to the differentiator circuit.

Comparison circuit 21.contains a tube 25 having a control electrode which is coupled to terminals 11 and 12 of the source of variable voltage 10. The anode of tube 25 is connected to a source of potential +13%. The cathode circuit includes resistor 26 in series with source of bias potential 42. Source of bias potential 42 acts to extend the range of operation of tube 25 for negative input signals. The cathode of tube 25, additionally, is coupled through a resistor 23 to the cathode of tube 29, a diode rectifier. The anode of tube 29 is connected to terminal 20 0f transformer 18. The cathode of tube 29 is also connected through condenser 30 to an input terminal of triggered pulse generator 31. Triggered pulse generator 31 may be a conventional blocking oscillator so that a detailed description thereof is unnecessary. If the output circuit of the comparison circuit 21 has insufficient magnitude, a pulse amplifier may be inserted between condenser 30 and the triggered pulse generator 31. Sampling circuit 32 is of the bridge rectifier type and comprises four diode vacuum tubes 33, 34,. 35', and 36 arranged ina conventional bridge rectifier circuit. One pair of diagonal terminals of this bridge circuit is cone nected between terminal 20 of transformer 18 and terminal 43. The other pair of diagonal terminals of the bridge is connected to an actuating circuit comprisingthe series combination of the output terminals of the triggered pulse generator 31 and a source of bias potential 38. V

Smoothing circuit 40 comprises a passive network of inductances and capacitances of suitable value for developing a signal whose amplitude is proportional to the amplitude of a signal applied to the input terminals. Terminals 41, 41 of the smoothing circuit 40 are connected across the series combination of condenser 13, resistor 26, and battery 42 from which the input signal to smoothing circuit 40 is derived. f .The diiferentiator further includes a circliit for ,de 'rI- mining the difference between the instantaneous value of the first voltage and the voltage of the storage device,

Such circuit consists of the series connection of source of variable voltage and condenser 13 with respect to terminals 41, 41.

Direct-current amplifier 39 represents a means for utilizing the difference voltage between the source of variable voltage and the voltage of condenser 13. The input terminals of direct-current amplifier 39 are coupled to the output terminals of the smoothing circuit 40'.

minal of transformer 18 is applied to one of the 7 inputs of comparison circuit 21.

Simultaneously, the variable voltage source 10, the voltage of which is representative of the variable parameter to be differentiated, applies a signal to a second input of comparison circuit 21.

When the amplitude of the sinusoidal signal from terminal 20 and appearing on the anode of tube 29'becomes equal to the instantaneous value of the variable parameter appearing on the cathode of tube 25, comparison circuit 21 develops a trigger signal which is applied through condenser 30 to triggered pulse generator 31. Triggered pulse generator 31 is activated and, in turn, develops an output pulse which is applied to sampling circuit 32. The pulse from generator 31 is of the proper polarity and magnitude to overcome the bias voltage developed by battery 38. Accordingly, the bridge circuit including tubes 33, 34, 35, and 36 is actuated and rendered conductive for the duration of the applied pulse with the result that the connection between terminal 20 and terminal 14 is completed. As a consequence, condenser 13 charges up substantially instantaneously to the instantaneous value of the sinusoidal signal at terminal 20. The maximum charging period is equal to the sampling period a and is made small enough so that both the variable parameter and the sinusoidal signal may be considered constant for the duration of the sampling period. In addition, the time constant developed by condenser 13 and the series impedance of sampling circuit 32, in a conductive state, is sufiiciently small so that it is virtually ineffective in delaying the accumulation of voltage on condenser 13. At the conclusion of sampling period oz, determined by the duration of the pulse from triggered pulse generator 31, sampling circuit 32 is disabled, storing the newly accumulated voltage in condenser 13.

At a subsequent time, corresponding to the next period ofthe sinusoidal signal, the amplitudes of the joint in puts to comparison circuit 21 are again equal, thereby establishing the required conditions for repeating the aforementioned cycle.

The, manner in which the Fig. .1 diiferentiator is utilized to perform differentiation will be more fully apparent from the following mathematical analysis of its operation as explained with reference to the curves of Figs. 2a and, 2b.

It is desired that equations of the following type be solved by the Fig. 1 embodiment:

dE(t) v The solution of the equation may be approximated by:

AE(t) l E. (2)

This approximation may be as accurate as desired. bymaking At as small as possible, having it approachzeroin the'limit.

' period a.

One means for obtaining the solution is to store for use at a time t the value of E(t) at time (t,,-At). It unit time At is taken as (t -t then E(t )E(t is the change A(t) which occurs in time At and, therefore, is equal to the approximation of the derivative represented in Equation 2 and very nearly equal to the derivative of Equation 1. Thus the source of variable voltage 10- of Fig. 1 may represent the source of the variable parameter B(t) providing instantaneous values of E(t). One cycle of the differentiator represented by Fig. 1 may be equal in time to At. Therefore, a value of E(t) for each At may be determined.

The operation of the embodiment represented in Fig. 1 will be discussed in detail with reference to the curves in Figs. 2a and 2b. E(t) in Fig. 2a represents an assumed variable parameter It is understood, however, that any function may be used. As represented, E(t) is the voltage appearing at terminal 11 of source of variable voltage 10 with respect to terminal '15 of condenser 13. The curve V represents the sinusoidal signal derived from sine-wave generator 17 and appears at terminal 2%) with respect to terminal 15 of condenser 13. The peak amplitude of V must exceed the values of E( t) in the interval of interest. In addition, the period At between samplings, determined by the frequency of V must be less than the time required for any appreciable change in Under the latter condition, E( t) may be considered substantially linear during the period At between samplings. Curve E represents the voltage stored in condenser 13 as a function of time. The curve represented in Fig. 2b as V denotes the difference voltage between the variable parameter E( t) and the voltage E stored in condenser 13.

The operation to obtain the derivative of E(t) will now be explained in detail. At the start of the problem, the voltage V at terminal 20 of transformer 18 starts to rise in a sinusoidal fashion. At the instant t during its excursion the amplitude of V equals E(t Comparison circuit 21 detects this equality and, in the manner heretofore described, causes terminal 14 of condenser 13 to be connected to terminal 20 of transformer 18 fora Consequently, condenser 13 charges to the voltage E(t and retains this voltage when sampling circuit 32 is disabled. Voltage V continues in a positive direction towards a maximum value and completes its cycle. The recovery characteristics of comparison circuit 21 in conjunction with triggered pulse generator 31 are adjusted to assure that only a single enabling signal to sampling circuit 32 is developed during a cycle of V During the next cycle of V it reaches the value E(t at time t and the equality is detected by comparison circuit 21. Condenser 13 is once more connected across theoutput o-f transformer 18 and charges to E(t The voltage E across condenser 13 always lags E(t) between samplings and the voltage between terminal 14 and the cathode of tube 25 is the saw-tooth curve V shown in Fig. 2b. It is apparent from Fig. 2b that the amplitude of the saw-tooth at the end of a period At represents the differenceEU --E(t or AE(t).

As heretofore discussed, E(t) in the period between sampling periods is deemed to be linear by virtue of choosing the frequency of V to be greater than the highest frequency of interest in the Fourier transform of E(t) and the saw-tooth is derived from this relationship. A voltage representative of AE(t) is obtained by applying the saw-tooth representing E(t E(t,, to smoothing circuit 40. Since the saW-toothis linear, its average value is proportional to its peak value. The smoothing circuit 40 should, therefore, average the individual saw-tooth without smoothing out dE-(r) am ss S The requirement is that the cutoff frequency of 40 should be lower than the frequency of V but higher than the highest frequency of interest in the Fourier transform of dE(r) dz The output of 40 is, therefore, the average value of the saw-tooth Which is proportional to the peak value of the sawtooth or proportional to dE(r) dz The time constant of smoothing circuit 40 must be large enough to permit the smoothing circuit 40 to retain values of AE(t) between sampling periods but it must be small enough to allow the output voltage from smoothing circuit 40 to follow the long term variations in AEU). Accordingly, for example, if the period of V is one-fourth the period of the highest frequency component of E), the time constant of smoothing circuit 40 would be between these values. Under the condition just discussed, a time constant equal to one-third of the period of the highest frequency component of E(t) will be suitable. The output voltage from smoothing circuit 40 is, therefore, proportional to the difference voltage E(t )E(t or k[AE(t)]. k is a constant of proportionality determined by the relation of the time constant of smoothing circuit 41 and the sampling period. This voltage is applied to direct-current amplifier 39 whose gain is adjusted to multiply the voltage represent ing k[AE(t)] by a constant inversely proportional to At(k). Consequently, the voltage at the output of direct-current amplifier 39 is equal to the approximation AE(t) and very nearly'equal to the derivative dE(t) dt The periods At between sampling periods are not equal since the time at which the sampling period is initiated is related to the amplitude of E(t) by virtue of the required finite rise time of voltage V The difference in the At intervals may bemade as small as desired by making the voltage V rise steeply. The accuracy of the computation by the present invention with respect to the derivative of Equation 1 is further governed by the period of V The period of V must be less than the time required for a change in value of dE(t) dt in order to obtain a representative sampling of every fluctuation of E(t).

While there has been described What is at present considered to be the preferred embodiment of this invention, it will be obvious to those skilled in the art that various changes and modifications may be made therein without departing from the invention, and it is, therefore, aimed to cover all such changes and modifications as fall within the true spirit and scope of the invention.

What is claimed is:

1. An electrical differentiator comprising: circuit means for supplying a first voltage representing a function to be differentiated; a storage device; a circuit for substantially periodically and instantaneously applying said voltage to the storage device; a circuit for determining the difference between the instantaneous value of the first voltage and the voltage of the storage device; and means for utilizing said difference as a representation of the derivative of said function.

2. An electrical difierentiator comprising: circuit means for supplying a first voltage representing a function to be differentiated; a condenser; a circuit for substantially periodically and instantaneously applying said voltage to the condenser; a circuit for determining the difference between the instantaneous value of the first voltage and the voltage of the condenser; and means for utilizing said difierence as a representation of the derivative of said function.

3. An electrical differentiator comprising: circuit means for supplying a first voltage representing a function to be differentiated; a storage device; a circuit for substantially periodically and instantaneously applying said voltage to the storage device; a circuit including the circuit means for supplying the first voltage and storage device in series opposition for determining the difference between the instantaneous value of the first voltage and the voltage of the storage device; and means for utilizing said difference as a representation of the derivative of said function.

4. An electrical diiferentiator comprising: circuit means for supplying a first voltage representing a function to be differentiated; a condenser; a circuit for substantially periodically and instantaneously applying said voltage to the condenser; a circuit including the circuit means for supplying the first voltage and condenser in series opposition for determining the difference between the instantaneous value of the first voltage and the voltage of the condenser; and means for utilizing said diiference as a representation of the derivative of said function.

5. An electrical differentiator comprising: circuit means for supplying a first voltage representing a function of time to be differentiated; a storage device; a circuit for substantially periodically and instantaneously applying said voltage to the storage device; a circuit for determining the difference between the instantaneous value of the first voltage and the voltage of the storage device; and means for utilizing said difference as a representation of the derivative of said function with respect to time.

6. An electrical differentiator comprising: circuit means for supplying a first voltage representing a function to be differentiated; a storage device; a circuit for substantially periodically and instantaneously applying said voltage to the storage device; a circuit for developing a saw-tooth voltage constituting the difference between the instantaneous value of the first voltage and the voltage of the storage device; a circuit for developing a voltage proportional to the peak amplitude of the saw-tooth voltage; and means for utilizing said proportional voltage as a representation of the derivative of said function.

7. An electrical differentiator comprising: circuit means for supplying a first voltage representing a function to be differentiated; a storage device; a circuit for substantially periodically and instantaneously applying said voltage to the storage device; a circuit for developing a saw-tooth voltage constituting the difference between the instantaneous value of the first voltage and the voltage of the storage device; a passive network for developing a voltage proportional to the peak amplitude of the saw-tooth voltage; and means for utilizing said proportional voltage as a representation of the derivative of said function.

8. An electrical diiferentiator comprising: means for supplying a periodic voltage; circuit means for supplying a first voltage representing a function to be differentiated; a storage device; comparison means for instantaneously comparing the periodic voltage with the first voltage; a circuit actuated by the comparison means for instantaneously applying said first voltage to the storage device when said periodic and first voltages become equal during the cycle of the periodic voltage; a circuit for determining the difference between the instantaneous value of the first voltage and the voltage of the storage device; and means for utilizing said difference as a representation of the derivative ofsaid function.

9. An electrical diiferentiator comprising: means for supplying a periodic voltage; circuit means for supplying a first voltage representing a function to be differentiated;

a condenser; comparison means for instantaneously comparing the periodic voltage with the first voltage; a circuit actuated by the comparison means for instantaneously applying said first voltage to the condenser when said periodic and first voltages become equal during the cycle of the periodic voltage; a circuit for determining the difference between the instantaneous value of the first voltage and the voltage of the condenser; and means for utilizing said difference as a representation of the derivative of said function.

10. An electrical diiferentiator comprising: means for supplying a periodic voltage; circuit means for supplying a first voltage representing a function to be difierentiated;

a storage device; comparison means for instantaneously comparing the periodic voltage with the first voltage; a circuit actuated by the comparison means for instantaneously applying said first voltage to the storage device when said periodic and first voltages become equal during the cycle of the periodic voltage; a circuit including the circuit means for supplying the first voltage and storage device in series opposition for determining the difference between the instantaneous value of the first voltage and the voltage of the storage device; and means for utilizing said difference as a representation of the derivative of said function.

11. An electrical differentiator comprising: means for supplying a periodic voltage; circuit means for supplying a first voltage representing a function to be differentiated; a condenser; comparison means for instantaneously comparing the periodic voltage with the first voltage; a circuit actuated by the comparison means for instantaneously applying said first voltage to the condenser when said periodic and first voltages become equal during the cycle of the periodic voltage; a circuit insluding the circuit means for supplying the first voltage and condenser in series opposition for determining the difierence between the instantaneous value of the first voltage and the voltage of the condenser; and means for utilizing said difference as a representation of the derivative of said function.

12. An electrical differentiator comprising: means for supplying a periodic voltage; circuit means for supplying a first voltage representing a function of time to be differentiated; a storage device; comparison means for instantaneously comparing the periodic voltage with the first voltage; a circuit actuated by the comparison means for instantaneously applying said first voltage to the storage device when said periodic and first voltages become equal during the cycle of the periodic voltage; a circuit for determining the difference between the instantaneous value of the first voltage and the voltage of the storage device; and means for utilizing said difference as a representation of the derivative of said function with respect to time.

13. An electrical difierentiator comprising: means for supplying a periodic voltage; circuit means for supplying a first voltage representing a function to be differentiated; a storage device; comparison means for instantaneously comparing the periodic voltage with the first voltage; a switch closed by the comparison means for instantaneously applying said first voltage to the storage device when said periodic and first voltages become equal during the cycle of the periodic voltage; a circuit for determining the difference between the instantaneous value of the first voltage and the voltage of the storage device; and means for utilizing said difference as a representation of the derivative of said function.

14. An electrical difierentiator comprising: means for supplying a periodic voltage; circuit means for supplyinga first voltage representing a function to be differentiated; a condenser; a comparator for instantaneously comparing the periodic voltage with the first voltage; a switching circuit actuated by the comparator for instantaneously applying said first voltage to the condenser when said periodic and first voltages become equal during the cycle of the periodic voltage; a circuit for determining the difference between the instantaneous value of the first voltage and the voltage of the condenser; and means for utilizing said difference as a representation of the derivative of said function.

15. An electrical difierentiator comprising: means for supplying a periodic voltage; circuit means for supplying a first voltage representing a function to be differentiated; a storage device; comparison means for instantaneously comparing the periodic voltage with the first voltage; a circuit actuated by the comparison means for instantaneously applying said first voltage to the storage device when said periodic and first voltages become equal during the cycle of the periodic voltage; a circuit for developing a saw-tooth voltage constituting the difference between the instantaneous value of the first voltage and the voltage of the storage device; a circuit for developing a voltage proportional to the peak amplitude of the sawtooth voltage; and means for utilizing said proportional voltage as a representation of the derivative of said function.

16. An electrical differentiator comprising: means for supplying a periodic voltage; circuit means for supplying a first voltage representing a function of time to be diiferentiated; a condenser; a comparator for instantaneously comparing the periodic voltage with the first voltage; a switch closed by the comparator for instantaneously applying said first voltage to the condenser when said periodic and first voltages'become equal during the cycle of the periodic voltage; a circuit including the circuit means for supplying the first voltage and condenser in series opposition for developing a saw-tooth voltage constituting the diiference between the instantaneous value of the first voltage and the voltage of the condenser; a circuit for developing a voltage proportional to the peak amplitude of the saw-tooth voltage; and means for utilizing said proportional voltage as a representation of the derivative of said function with respect to time.

17. An electrical diiferentiator comprising: means for supplying a periodic voltage; circuit means for supplying a first voltage representing a function to be differentiated; a storage device; comparison means for instantaneously comparing the periodic voltage with the first voltage; a circuit actuated by the comparison means for instantaneously applying said voltage to the storage device when said periodic and first voltages become equal during the cycle of the periodic voltage; a circuit for developing a saw-tooth voltage constituting the difierence between the instantaneous value of the first voltage and the voltage of the storage device; a passive circuit for developing a voltage proportional to the peak amplitude of the sawtooth voltage; and means for utilizing said proportional voltage as a representation of the derivative of said function.

References Cited in the file of this patent UNITED STATES PATENTS 2,437,313 Bedford Mar. 9, 1948 2,794,173 Rarney May 28, 1957 2,822,978 Donovan Feb. 11, 1958 th-AG n= I 

